Do black holes have charges? If so, how would they be measured? Also, does electricity behave the same way? Black holes affect photons, which are carriers of EM radiation, so do black holes have any effect on the electric force?
Our current understanding suggests that black holes can have electric charge, and that in addition to mass and angular momentum, these are the only ways that black holes can have distinguishable physical properties. This is a result of the famous no-hair theorem, although keep in mind that no definitive proof of this theorem yet exists.
As this wikipedia entry on charged black holes points out, though, the electric charge of a black hole is generally expected to be quite small.
Far away from the black hole, you could in principle measure the electric charge in the same you measure any other electric field, say with an electrometer, although this it seems like it would be quite difficult to make a sensitive enough measurement. As you get closer to the black hole, the geometry of the field lines will be affected by the gravity of the black hole, as reflected in its space-time metric. Again, putting practicalities aside, it would also be possible in principle to measure the charge of the black hole by measuring its gravitational signature via the metric and solving for the charge, if one already knows its mass.
These are two separate questions. It's better if you don't try to combine two questions into one.
In answer to the first question, yes, a black hole can have a measurable charge. You measure it the same way you'd measure any other charge. This is all purely theoretical, however. Many real-life black holes have been observed and characterized by their masses, but there is no realistic chance that any black hole will ever have enough charge to be measurable.
The way we know that black holes can be charged is that there exist solutions to the Einstein field equations such as the Reissner–Nordström metric, in which a charged test particle experiences a force not experienced by a neutral test particle. The no-hair theorems for black holes explicitly allow electric charge as an observable type of "hair" that it is possible for a black hole to have.
In answer to the second question, Maxwell's equations can be expressed in tensor form, and in this form, they work equally well in general relativity.